Some Problems in Applied Nonlinear Partial Differential Equations

Iowa State University, Ames IA

Investigators

Abstract

The purpose of the project is to probe the dissipation mechanism in viscous approximation of multidimensional compressible potential flow by studying the regularity property, positivity of density, and dynamics of convective singularity surfaces, to justify the relation between variety of plate theories and three dimensional elasticity theory by deriving the asymptotic limit of three dimensional nonlinear elasticity, to deepen the understanding of optimal transport mechanism in non-turbulent semi-geostrophic equations, to investigate evolution of interface in multi-phase crystal growth in material science, stochastic solutions of Hamilton-Jacobi equations which model many interesting physical processes such as front propagation in combustion. Rigorous mathematics in the study of the nonlinear partial differential equtions addressed in the project make it possible both to conduct stable numerical computation in, and to understand the qualitative features of the phenomena addressed above. One of our objectives is to study viscous oompressible potential flow and semi-geostrophic flow. The advance in the knowledge of global large solutions of mathematical equations has important impact on our fundamental understanding of air flow. The second objective is to analyze the front propagation of crystal growth and thin-film blistering in the semiconducting materials from which computing devices are manufactured. The third objective of this project is the training of graduate students and postdoctoral fellows in the mathematical analysis of applied problems.